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293
Semidefinite Programming Relaxations for Semialgebraic Problems
, 2001
"... A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The mai ..."
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Cited by 310 (22 self)
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A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.
The Convex Geometry of Linear Inverse Problems
, 2010
"... In applications throughout science and engineering one is often faced with the challenge of solving an illposed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constr ..."
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Cited by 170 (17 self)
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In applications throughout science and engineering one is often faced with the challenge of solving an illposed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include wellstudied cases such as sparse vectors (e.g., signal processing, statistics) and lowrank matrices (e.g., control, statistics), as well as several others including sums of a few permutations matrices (e.g., ranked elections, multiobject tracking), lowrank tensors (e.g., computer vision, neuroscience), orthogonal matrices (e.g., machine learning), and atomic measures (e.g., system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial
Flag algebras
 Journal of Symbolic Logic
"... Abstract. Asymptotic extremal combinatorics deals with questions that in the language of model theory can be restated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a random ..."
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Cited by 81 (8 self)
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Abstract. Asymptotic extremal combinatorics deals with questions that in the language of model theory can be restated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a randomly chosen submodel of N with M  elements is isomorphic to M. Which asymptotic relations exist between the quantities p(M1, N),..., p(Mh, N), where M1,..., Mh are fixed “template ” models and N  grows to infinity? In this paper we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density. §1. Introduction. A substantial part of modern extremal combinatorics (which will be called here asymptotic extremal combinatorics) studies densities with which some “template ” combinatorial structures may or may not appear in unknown (large) structures of the same type1. As a typical example, let Gn be a
Minimizing polynomial functions
 PROCEEDINGS OF THE DIMACS WORKSHOP ON ALGORITHMIC AND QUANTITATIVE ASPECTS OF REAL ALGEBRAIC GEOMETRY IN MATHEMATICS AND COMPUTER SCIENCE
, 2003
"... We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gröbner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves su ..."
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Cited by 58 (3 self)
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We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gröbner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves sums of squares and semidefinite programming. This opens up the possibility of using semidefinite programming relaxations arising from the Positivstellensatz for a wide range of computational problems in real algebraic geometry.
Linear time logic control of discretetime linear systems
 IEEE Transactions on Automatic Control
, 2006
"... Abstract. The control of complex systems poses new challenges that fall beyond the traditional methods of control theory. One of these challenges is given by the need to control, coordinate and synchronize the operation of several interacting submodules within a system. The desired objectives are no ..."
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Cited by 58 (4 self)
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Abstract. The control of complex systems poses new challenges that fall beyond the traditional methods of control theory. One of these challenges is given by the need to control, coordinate and synchronize the operation of several interacting submodules within a system. The desired objectives are no longer captured by usual control specifications such as stabilization or output regulation. Instead, we consider specifications given by Linear Temporal Logic (LTL) formulas. We show that existence of controllers for discretetime controllable linear systems and LTL specifications can be decided and that such controllers can be effectively computed. The closedloop system is of hybrid nature, combining the original continuous dynamics with the automatically synthesized switching logic required to enforce the specification. 1.
Alternating minimization and projection methods for nonconvex problems
 0801.1780v2[math.oc], arXiv
, 2008
"... Abstract We study the convergence properties of alternating proximal minimization algorithms for (nonconvex) functions of the following type: L(x,y) = f(x) + Q(x,y) + g(y) where f: R n → R∪{+∞} and g: R m → R∪{+∞} are proper lower semicontinuous functions and Q: R n ×R m → R is a smooth C 1 (finite ..."
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Cited by 58 (2 self)
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Abstract We study the convergence properties of alternating proximal minimization algorithms for (nonconvex) functions of the following type: L(x,y) = f(x) + Q(x,y) + g(y) where f: R n → R∪{+∞} and g: R m → R∪{+∞} are proper lower semicontinuous functions and Q: R n ×R m → R is a smooth C 1 (finite valued) function which couples the variables x and y. The algorithm is defined by: (x0, y0) ∈ R n × R m given, (xk, yk) → (xk+1, yk) → (xk+1, yk+1)
A survey of the Slemma
 SIAM Review
"... Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as ..."
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Cited by 56 (1 self)
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Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the Slemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. Slemma, Sprocedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities
Computational and Statistical Tradeoffs via Convex Relaxation
, 2012
"... In modern data analysis, one is frequently faced with statistical inference problems involving massive datasets. Processing such large datasets is usually viewed as a substantial computational challenge. However, if data are a statistician’s main resource then access to more data should be viewed as ..."
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Cited by 40 (1 self)
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In modern data analysis, one is frequently faced with statistical inference problems involving massive datasets. Processing such large datasets is usually viewed as a substantial computational challenge. However, if data are a statistician’s main resource then access to more data should be viewed as an asset rather than as a burden. In this paper we describe a computational framework based on convex relaxation to reduce the computational complexity of an inference procedure when one has access to increasingly larger datasets. Convex relaxation techniques have been widely used in theoretical computer science as they give tractable approximation algorithms to many computationally intractable tasks. We demonstrate the efficacy of this methodology in statistical estimation in providing concrete timedata tradeoffs in a class of denoising problems. Thus, convex relaxation offers a principled approach to exploit the statistical gains from larger datasets to reduce the runtime of inference algorithms.